Determinant Of 2AB: Step-by-Step Calculation

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Hey guys! Let's dive into a super interesting problem today: figuring out the determinant of a matrix expression. We've got square matrices A and B, both of order 3, and we know their determinants: |A| = -9 and |B| = -8. The mission? Find |2AB|. Sounds like a fun math adventure, right? So, let's break it down and make it crystal clear. Buckle up, because we're about to unravel the magic behind determinants!

Understanding the Basics of Determinants

Before we jump into the calculation, let's make sure we're all on the same page about what a determinant actually is. In simple terms, the determinant of a matrix is a special number that can be computed from the elements of a square matrix. It gives us valuable information about the matrix, such as whether the matrix is invertible (non-singular) or not. If the determinant is non-zero, the matrix is invertible. If it's zero, the matrix is singular, meaning it doesn't have an inverse.

For a 2x2 matrix, say:

| a  b |
| c  d |

The determinant is calculated as (ad - bc). Things get a bit more intricate for larger matrices like 3x3 or higher, but the underlying principle remains the same: it’s a specific formula applied to the matrix elements.

The determinant has some cool properties that are super helpful in calculations. Here are a couple of key ones we'll use today:

  1. Determinant of a Product: The determinant of the product of two matrices is the product of their determinants. Mathematically, this means |AB| = |A| * |B|.
  2. Determinant of a Scalar Multiple: If you multiply a matrix by a scalar (just a regular number), the determinant changes in a predictable way. For a matrix A of order n and a scalar k, |kA| = kn|A|. This is crucial because it tells us how the determinant scales when we multiply a matrix by a constant.

Why is this important? Well, determinants pop up all over the place in linear algebra and its applications. They’re used in solving systems of linear equations (think Cramer's Rule), finding eigenvalues, and even in geometry to calculate areas and volumes. So, understanding determinants is like unlocking a superpower in the world of matrices!

Applying Determinant Properties to Solve |2AB|

Now, let's roll up our sleeves and get to the heart of the problem: finding |2AB|. We know |A| = -9 and |B| = -8, and we need to use the properties we just discussed to figure out the determinant of the matrix 2AB. Remember, the order of both matrices A and B is 3, which is super important for our calculations.

First, let's use the property that the determinant of a product is the product of the determinants. This means:

|2AB| = |2A| * |B|

This step breaks down the problem into smaller, more manageable chunks. We now need to find |2A| and we already know |B| = -8. So, let's focus on |2A|.

Here's where the second key property comes into play: the determinant of a scalar multiple. Remember, for a matrix A of order n and a scalar k, |kA| = kn|A|. In our case, k = 2 and n = 3 (since A is a 3x3 matrix). So, we have:

|2A| = 23 * |A|

This is where the order of the matrix really matters. The exponent 3 comes directly from the fact that A is a 3x3 matrix. If A were a 2x2 matrix, we’d have 22, and so on.

Now, we can calculate 23, which is 8. And we know |A| = -9, so:

|2A| = 8 * (-9) = -72

Alright, we're on the home stretch! We've found |2A| = -72, and we know |B| = -8. Now we just need to multiply these together:

|2AB| = |2A| * |B| = (-72) * (-8)

Step-by-Step Calculation of |2AB|

Let's recap the steps we've taken so far. This will help solidify the process and make sure we haven't missed anything. Breaking down the calculation into steps makes it easier to follow and less intimidating, especially when dealing with more complex problems.

  1. Recognize the Problem: We need to find the determinant of the matrix 2AB, given |A| = -9 and |B| = -8, where A and B are 3x3 matrices.
  2. Apply the Product Rule: Use the property |AB| = |A| * |B| to rewrite |2AB| as |2A| * |B|. This allows us to deal with the scalar multiple separately.
  3. Handle the Scalar Multiple: Use the property |kA| = kn|A|, where n is the order of the matrix. For |2A|, this becomes 23 * |A|.
  4. Calculate |2A|: Since |A| = -9 and 23 = 8, we get |2A| = 8 * (-9) = -72.
  5. Multiply the Determinants: Now we multiply |2A| and |B|: |2AB| = (-72) * (-8).
  6. Perform the Final Calculation: Multiply -72 by -8 to get the final answer.

Now let's complete that final calculation:

|2AB| = (-72) * (-8) = 576

And there we have it! The determinant of 2AB is 576.

The Final Answer: |2AB| = 576

So, after breaking down the problem step-by-step and applying the properties of determinants, we've found that |2AB| = 576. Isn’t that awesome? We started with some given information and, using the magic of linear algebra, arrived at a clear, concrete answer.

To recap, we used two key properties:

  • |AB| = |A| * |B| (the determinant of a product)
  • |kA| = kn|A| (the determinant of a scalar multiple)

These properties are incredibly useful in simplifying determinant calculations, especially when dealing with matrix expressions. Remember, the order of the matrix is crucial when dealing with scalar multiples, as it affects the exponent we use.

This problem is a fantastic example of how understanding the fundamental properties of determinants can make complex calculations manageable. It’s like having a set of tools that allow you to dissect and solve problems efficiently. So, keep these properties in mind, and you'll be well-equipped to tackle any determinant challenge that comes your way. Keep practicing, and you'll become a determinant pro in no time!

In conclusion, the determinant of 2AB, given |A| = -9 and |B| = -8 for 3x3 matrices, is 576. Great job, guys! We nailed it!